Let G be a graph and Ḡ be the complement of G. The complementary prism GḠ of G is the graph formed from the disjoint union of G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. For example, if G is a 5-cycle, then GḠ is the Petersen graph. In this paper we consider domination and total domination numbers of complementary prisms. For any graph G, max {γ(G), γ(Ḡ)} ≤ γ (Ḡ)and max {γt(G), γt(Ḡ)} ≤ γt (Gγ), where γ(G) and γt(G) denote the domination and total domination numbers of G, respectively. Among other results, we characterize the graphs G attaining these lower bounds.

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one that are γ t-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γt-critical graph for k≤8 and provide an example which shows that the maximum diameter is in general at least 5k/3 - O(1).

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Fricke, Haynes, Hedetniemi, Hedetniemi, and Laskar introduced the following concept. For a graph G = (V,E), let rho denote a property of interest concerning sets of vertices. A vertex u is rho-good if u is contained in a {minimum, maximum} rho-set in G and rho-bad if u is not contained in a rho-set. Let g denote the number of rho-good vertices and b denote the number of rho-bad vertices. A graph G is called rho-excellent if every vertex in V is rho-good, rho-commendable if g > b > 0, rho-fair if g = b, and rho-poor if g < b. In this thesis the property of interest is total domination. The total domination number, gammat, is the cardinality of a smallest total dominating set in a graph. We investigate gammat-excellent, gammat-commendable, gammat-fair, and gammat-poor graphs.

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.

The Queen’s domination problem has a long and rich history. The problem can be simply stated as: What is the minimum number of queens that can be placed on a chessboard so that all squares are attacked or occupied by a queen? The problem has been expanded to include not only the standard 8x8 board, but any rectangular m×n sized board. In this thesis, we consider both paired and total domination versions of this renowned problem.

We study the concept of strong equality of domination parameters. Let P1 and P2 be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P2 also has property P1. Let ψ1(G) and ψ2(G), respectively, denote the minimum cardinalities of sets with properties P1 and P2, respectively. Then ψ1(G) ≤ ψ2(G). If ψ1(G)=ψ2(G) and every ψ1(G)-set is also a ψ2(G)-set, then we say ψ1(G) strongly equals ψ2(G), written ψ1(G) = ψ2(G). We provide a constructive characterization of the trees T such that γ(T) = i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T) = γt(T), where γt(T) denotes the total domination number of T, is also presented.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.

A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph G of order n is at most ⌊n2/4⌋ and that the extremal graphs are the complete bipartite graphs K⌊n/2⌋,⌈n/2⌉. A graph is t-edge-critical, abbreviated 3tEC, if its total domination number is 3 and the addition of any edge decreases the total domination number. It is known that proving the Murty–Simon Conjecture is equivalent to proving that the number of edges in a 3tEC graph of order n is greater than ⌈n(n-2)/4⌉. We study a family F of 3tEC graphs of diameter 2 for which every pair of nonadjacent vertices dominates the graph. We show that the graphs in F are precisely the bull-free 3tEC graphs and that the number of edges in such graphs is at least ⌊(n2-4)/4⌋, proving the conjecture for this family. We characterize the extremal graphs, and conjecture that this improved bound is in fact a lower bound for all 3tEC graphs of diameter 2. Finally we slightly relax the requirement in the definition of F—instead of requiring that all pairs of nonadjacent vertices dominate to requiring that only most of these pairs dominate—and prove the Murty–Simon equivalent conjecture for these 3tEC graphs.

A set S of vertices in a graph G is a global dominating set of G if S simultaneously dominates both G and its complement Ḡ. The minimum cardinality of a global dominating set of G is the global domination number of G. We determine bounds on the global domination number of a graph and relationships between it and other domination related parameters.